Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2383 https://internationalpubls.com Reimagining Fixed Points: Exploring the Role of Occasionally Weakly Compatible Mappings in Fuzzy Metrics 1Priyanka Nigam, 2Sandhya Shukla 1Presidency University, Bengaluru, 560089, Karnataka, India. E-mail: priyanka.nigam@presidencyuniversity.in, priyanka_nigam01@yahoo.co.in 2University Institute of Technology, RGPV Bhopal, 462033, Madhya Pradesh, India E-mail: maths.sandhyashukla@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this work, we revisit the concept of fixed points by exploring the role of occasionally weakly compatible (OWC) mappings within the structure of fuzzy metric spaces. Building on the classical fixed point theory, we investigate new conditions under which tripled fixed points exist and are unique. By employing the framework of fuzzy metrics and leveraging the flexibility of OWC mappings, we establish a generalized tripled fixed point theorem that extends several known results. We also explore the concept of tripled fixed points for occasionally weakly compatible mappings within the framework of fuzzy metric spaces. We establish several novel tripled fixed-point theorems that extend existing results in this area. Additionally, to validate the applicability of our theorems, we provide detailed illustrative examples that demonstrate the effectiveness and relevance of the established results in fuzzy metric settings. These findings contribute to the broader understanding of fixed-point theory in fuzzy environments and open new avenues for future research in generalized metric spaces and their applications. Keywords: Occasionally weakly compatible mappings; tripled fixed point; fuzzy metric space. 2000 Mathematics Subject Classification: 47H10; 54H25. 1 Introduction Zadeh [14] defined Fuzzy sets. Kramosil and Michalek [7] introduced Fuzzy metric space, George and Veermani [3] modified the notion and gave a new notion with the help of continuous t-norms of fuzzy metric spaces. Many researchers have obtained common fixed point theorems for mappings satisfying different types of commutativity conditions. Fixed point theorems, involving four self-maps, began with the assumption that they are commuted. Sessa [10] weakened the condition of commutativity to that of pairwise weakly commuting. Jungck generalized the notion of weak commutativity to that of pairwise compatible [4] and then pairwise weakly compatible maps [5]. Jungck and Rhoades [6] introduced the concept of occasionally weakly compatible maps (owc). Some of the work cited in references [1], [2], [8], [9], [11], [12] and [13] is also significant. In this work, we introduce tripled fixed point for occasionally weakly compatible mappings in fuzzy metric space and also proved some tripled fixed-point theorem for occasionally weakly compatible mailto:priyanka.nigam@presidencyuniversity.in mailto:priyanka_nigam01@yahoo.co.in mailto:maths.sandhyashukla@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2384 https://internationalpubls.com mappings in fuzzy metric space. Our results extend and some recent results in literature. Some illustrative examples are offered to support our theorems. We have also given the diagram to demonstrate the viability and applicability of the result. 2 Preliminary Notes Definition 2.1 A fuzzy set A in X is a function with domain X and values in [0, 1]. Definition 2.2 A binary operation ∗∶ [0,1] [0,1]→ [0,1] is a continuous t-norm if ∗ is satisfying conditions: (i) ∗ is an commutative and associative; (ii) ∗is continuous; (iii) 𝑎 ∗ 1 = 𝑎 for all a [0,1]; (iv) 𝑎 ∗ 𝑏𝑐 ∗ 𝑑 whenever ca  𝑎𝑛𝑑 db  and 𝑎, 𝑏, 𝑐, 𝑑[0,1]. Definition 2.3 A 3-tuple (𝑋,𝑀,∗) is said to be a fuzzy metric space if X is an arbitrary set ,∗ is a continuous 𝑡 − 𝑛𝑜𝑟𝑚 and M is a fuzzy set on ( ) ,02X satisfying the following conditions, for all 𝑥, 𝑦, 𝑧  𝑋, 𝑠, 𝑡 > 0, (𝑖) 0),,( tyxM ; (𝑖𝑖) 1),,( =tyxM 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 yx = ; (𝑖𝑖𝑖) ),,( tyxM = ),,( txyM ; (𝑖𝑣) ),,( tyxM ∗ ),,( szyM ),,( stzxM + ; (𝑣) ]1,0(),0(:),,( →yxM is continuous. Then M is called a 𝑓𝑢𝑧𝑧𝑦 𝑚𝑒𝑡𝑟𝑖𝑐 on X. Then ),,( tyxM denotes the degree of nearness between x and y with respect to t. Example 2.4 Let ),( dX be a metric space. Denote 𝑎 ∗ 𝑏 = 𝑎𝑏 for all  1,0, ba and let dM be fuzzy sets on ( ) ,02X defined as follows: ),( yxdt t M d + = . Then (𝑋, 𝑀𝑑,∗) is a fuzzy metric space. Lemma 2.5Let (X, M,*) be a fuzzy metric space. If there exists )1,0(q such that M(x, y, qt)  M(x ,y ,t) for all x, y  X and t>0, then x = y. Definition 2.6Let X be a non-empty set. An element (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 is called a tripled fixed point of a given mapping 𝑓: 𝑋 × 𝑋 × 𝑋 → 𝑋 if 𝑥 = 𝑓(𝑥, 𝑦, 𝑧), 𝑦 = 𝑓(𝑦, 𝑧, 𝑥), 𝑧 = 𝑓(𝑧, 𝑥, 𝑦). Example 2.6.1Let 𝑋 = 𝑅 and 𝑆: 𝑋 × 𝑋 × 𝑋 → 𝑋is defined as 𝑆(𝑥, 𝑦, 𝑧) = 𝑥 + 𝑥𝑦 + 𝑥𝑧 then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2385 https://internationalpubls.com 𝑆(1,0,0) = 1, 𝑆(0,1,0) = 0, 𝑆(0,0,1) = 0 Then (1,0,0),(0,1,0) and (0,0,1) are tripled fixed point. Definition 2.7 An element 𝑥 ∈ 𝑋 is called a common tripled fixed point of the mappings 𝑓: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 if 𝑥 = 𝑓(𝑥, 𝑥, 𝑥) = 𝑔(𝑥). Definition 2.8 An element (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 is called a tripled coincidence point of a mapping 𝑓: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 if 𝑔𝑥 = 𝑓(𝑥, 𝑦, 𝑧), 𝑔𝑦 = 𝑓(𝑦, 𝑧, 𝑥), 𝑔𝑧 = 𝑓(𝑧, 𝑥, 𝑦) in this case (𝑔𝑥, 𝑔𝑦, 𝑔𝑧) is called a tripled point of coincidence. Definition 2.9 let 𝑓 ∶ 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔 ∶ 𝑋 → 𝑋 be two mappings. 𝑓 𝑎𝑛𝑑 𝑔 are said to be weakly compatible if they commute at their a tripled coincidence point, i.e., if (𝑥, 𝑦, 𝑧) is a tripled coincidence point of 𝑔 𝑎𝑛𝑑 𝑓, then 𝑔𝑓(𝑥, 𝑦, 𝑧) = 𝑓(𝑔(𝑥), 𝑔(𝑦), 𝑔(𝑧)). Example 2.9.1Let S: X × X × X → X& 𝑇: X → X be defined by S(x, y, z) = x + xy + xz 𝑇(x) = { 0, if x ≠ 1; 1, ifx = 1. Here, (1,0,0), (0,1,0) and (0,0,1) are triple coincidence points of S and T at which (S,T) commute. So, S and T are weakly compatible. Example 2.9.2Let S: X × X × X → X& 𝑇: X → X be defined by S(x, y, z) = xyz 𝑇(x) = { 0, if 0 ≤ x ≤ 1; 1, ifx ≥ 1. So, S and T are weakly compatible at (0, 0, 0) and (1, 1, 1). Definition 2.10The mappings𝑓: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔:𝑋 → 𝑋 of a set X are occasionally weakly compatible(𝑜𝑤𝑐)iff there is a point (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋 which is a coincidence point of f and g at which f and g commute i.e. (𝑓, 𝑔) are occasionally weakly compatible maps iff 𝑓(𝑥, 𝑦, 𝑧) = 𝑔(𝑥), 𝑓(𝑦, 𝑧, 𝑥) = 𝑔(𝑦), 𝑓(𝑧, 𝑥, 𝑦) = 𝑔𝑧 implies 𝑔𝑓(𝑥, 𝑦, 𝑧) = 𝑓(𝑔𝑥, 𝑔𝑦, 𝑔𝑧), 𝑔𝑓(𝑦, 𝑧, 𝑥) = 𝑓(𝑔𝑦, 𝑔𝑧, 𝑔𝑥), 𝑔𝑓(𝑧, 𝑥, 𝑦) = 𝑓(𝑔𝑧, 𝑔𝑥, 𝑔𝑦) for (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋. Example 2.10.1 Let (X, ℱ,∗) be a fuzzy metric space, where X = [0,1] with 𝑎 ∗ 𝑏 = min{𝑎, 𝑏}and M(x, y, t) = { t t + |x − y| , if t > 0; 0, if t = 0. Let f: X × X × X → X& 𝑔: X → X be defined by f(x, y, z) = 2x + 2y + z 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2386 https://internationalpubls.com g(x) = { x, if 0 ≤ x < 1; 5 2 , ifx ≥ 1. Here, (0,0,0) and (1,1,1) are two coincidence points of f and g. That isf(0,0,0) = 0 = g(0), f(1,1,1) = 1 = g(1)butgf(0,0,0) = 0 = f(g0, g0, g0), gf(1,1,1) ≠ f(g1, g1, g1). Thus f and g are owc but not weakly compatible. The Mesh Diagram for the given example is shown in Fig [2.1]. Fig [2.1] 3 Main Results Theorem: 3.1Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: (i) 𝑀𝑃(𝐴(𝑥, 𝑦, 𝑧), 𝐵(𝑢, 𝑣, 𝑤), 𝑞𝑡) ≥ 𝜑 [ 𝑎 𝑀𝑝(𝑆𝑥, 𝑇𝑢, 𝑡) + (1 − 𝑎) min {𝑀𝑝(𝐴(𝑥, 𝑦, 𝑧), 𝑆𝑥, 𝑡), 𝑀𝑝(𝐵(𝑢, 𝑣, 𝑤), 𝑇𝑢, 𝑡), 𝑀 𝑝 2(𝐴(𝑥, 𝑦, 𝑧), 𝑇𝑢, 𝑡). 𝑀 𝑝 2(𝐵(𝑢, 𝑣, 𝑤), 𝑆𝑥, 𝑡) , 1 2 [𝑀𝑝(𝐴(𝑥, 𝑦, 𝑧), 𝑆𝑥, 𝑡) + 𝑀𝑝(𝐵(𝑢, 𝑣, 𝑤), 𝑇𝑢, 𝑡)]} ] for all 𝑥, 𝑦, 𝑧, 𝑢, 𝑣, 𝑤 ∈ 𝑋, 0 ≤ 𝑎 ≤ 1, 𝑝 ≥ 1 and 𝜑: 𝑅+ → 𝑅+ such that 𝜑 is upper semi continuous, non- increasing and 𝜑(𝑡) > 𝑡 for any t>0. (ii) 𝑦 = 𝐵(𝑥, 𝑦, 𝑧) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2387 https://internationalpubls.com Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that𝐴(𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. Proof:Since the pairs (A,S) and (B,T) are owc so there are points 𝑎, 𝑏, 𝑐, 𝑎′, 𝑏′, 𝑐′ in X such that 𝐴(𝑎, 𝑏, 𝑐) = 𝑆𝑎, 𝐴(𝑏, 𝑐, 𝑎) = 𝑆𝑏, 𝐴(𝑐, 𝑎, 𝑏) = 𝑆𝑐and 𝐵(𝑎′, 𝑏′ , 𝑐′) = 𝑇𝑎′ , 𝐵(𝑏′, 𝑐′, 𝑎′) = 𝑇𝑏′, 𝐵(𝑐′, 𝑎′, 𝑏′) = 𝑇𝑐′ We claim that 𝑆𝑎 = 𝑇𝑎′ . If not, by inequality (𝑖) we get 𝑀𝑃(𝐴(𝑎, 𝑏, 𝑐), 𝐵(𝑎′, 𝑏′, 𝑐′), 𝑞𝑡) ≥ 𝜑 [ 𝑎 𝑀𝑝(𝑆𝑎, 𝑇𝑎′ , 𝑡) + (1 − 𝑎) min {𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡), 𝑀𝑝(𝐵(𝑎′, 𝑏′, 𝑐′), 𝑇𝑎′ , 𝑡), 𝑀 𝑝 2(𝐴(𝑎, 𝑏, 𝑐), 𝑇𝑎′ , 𝑡).𝑀 𝑝 2(𝐵(𝑎′, 𝑏′, 𝑐′), 𝑆𝑎, 𝑡) , 1 2 [𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡) + 𝑀𝑝(𝐵(𝑎′, 𝑏′, 𝑐′), 𝑇𝑎′, 𝑡)]} ] = 𝜑(𝑎 𝑀𝑝(𝑆𝑎, 𝑇𝑎′ , 𝑡) + (1 − 𝑎)min {1,1, 𝑀𝑝(𝑆𝑎, 𝑇𝑎′, 𝑡), 1} = 𝜑(𝑎 𝑀𝑝(𝑆𝑎, 𝑇𝑎′ , 𝑡) + (1 − 𝑎)min 𝑀𝑝(𝑆𝑎, 𝑇𝑎′, 𝑡) > 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑡) ⇒ 𝑆𝑎 = 𝑇𝑎′ Therefore 𝐴(𝑎, 𝑏, 𝑐) = 𝑇𝑎′ = 𝑆𝑎 = 𝐵(𝑎′ , 𝑏′ , 𝑐′) Similarly 𝐴(𝑏, 𝑐, 𝑎) = 𝑇𝑏′ = 𝑆𝑏 = 𝐵(𝑏′, 𝑐′, 𝑎′) 𝐴(𝑐, 𝑎, 𝑏) = 𝑇𝑐′ = 𝑆𝑐 = 𝐵(𝑐′, 𝑎′ , 𝑏′) Thus the pairs (𝐴, 𝑆)and(𝐵, 𝑇)have common coincidence points. Let 𝐴(𝑎, 𝑏, 𝑐) = 𝑇𝑎′ = 𝑆𝑎 = 𝐵(𝑎′, 𝑏′ , 𝑐′) = 𝑥 and 𝐴(𝑏, 𝑐, 𝑎) = 𝑇𝑏′ = 𝑆𝑏 = 𝐵(𝑏′, 𝑐′, 𝑎′) = 𝑦 𝐴(𝑐, 𝑎, 𝑏) = 𝑇𝑐′ = 𝑆𝑐 = 𝐵(𝑐′, 𝑎′ , 𝑏′) = 𝑧 Since(𝐴, 𝑆) and (𝐵, 𝑇) are owc So 𝑆𝑥 = 𝑆𝐴(𝑎, 𝑏, 𝑐) = 𝐴(𝑆𝑎, 𝑆𝑏, 𝑆𝑐) = 𝐴(𝑥, 𝑦, 𝑧) and 𝑆𝑦 = 𝑆𝐴(𝑏, 𝑐, 𝑎) = 𝐴(𝑆𝑏, 𝑆𝑐, 𝑆𝑎) = 𝐴(𝑦, 𝑧, 𝑥) 𝑆𝑧 = 𝑆𝐴(𝑐, 𝑎, 𝑏) = 𝐴(𝑆𝑐, 𝑆𝑎, 𝑆𝑏) = 𝐴(𝑧, 𝑥, 𝑦) Also 𝑇𝑥 = 𝑇𝐵(𝑎′, 𝑏′, 𝑐′) = 𝐵(𝑇𝑎′, 𝑇𝑏′, 𝑇𝑐′) = 𝐵(𝑥, 𝑦, 𝑧) 𝑇𝑧 = 𝑇𝐵(𝑐′, 𝑎′, 𝑏′) = 𝐵(𝑇𝑐′, 𝑇𝑎′, 𝑇𝑏′) = 𝐵(𝑧, 𝑥, 𝑦) Next we show that 𝑥 = 𝑦 = 𝑧, for this putting 𝑥 = 𝑎 , 𝑦 = 𝑏, 𝑧 = 𝑐, 𝑢 = 𝑏′ , 𝑣 = 𝑐′,𝑤 = 𝑎′ in (i), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2388 https://internationalpubls.com 𝑀𝑃(𝐴(𝑎, 𝑏, 𝑐), 𝐵(𝑏′, 𝑐′, 𝑎′), 𝑞𝑡) ≥ 𝜑 [ 𝑎 𝑀𝑝(𝑆𝑎, 𝑇𝑏′, 𝑡) + (1 − 𝑎) min {𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡), 𝑀𝑝(𝐵(𝑏′, 𝑐′, 𝑎′), 𝑇𝑏′, 𝑡),𝑀 𝑝 2(𝐴(𝑎, 𝑏, 𝑐), 𝑇𝑏′, 𝑡).𝑀 𝑝 2(𝐵(𝑏′, 𝑐′, 𝑎′), 𝑆𝑎, 𝑡) , 1 2 [𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡) + 𝑀𝑝(𝐵(𝑏′, 𝑐′, 𝑎′), 𝑇𝑏′, 𝑡)]} ] 𝑀𝑝(𝑥, 𝑦, 𝑞𝑡) ≥ 𝜑(𝑎 𝑀𝑝(𝑥, 𝑦, 𝑡) + (1 − 𝑎)𝑚𝑖𝑛{1,1,𝑀𝑝(𝑥, 𝑦, 𝑡), 1} = 𝜑𝑀𝑝(𝑥, 𝑦, 𝑡) > 𝑀(𝑥, 𝑦, 𝑡) ⟹ 𝑥 = 𝑦 Again putting 𝑥 = 𝑎 , 𝑦 = 𝑏, 𝑧 = 𝑐, 𝑢 = 𝑐′ , 𝑣 = 𝑎′,𝑤 = 𝑏′ in (i), 𝑀𝑃(𝐴(𝑎, 𝑏, 𝑐), 𝐵(𝑐′, 𝑎′ , 𝑏′), 𝑞𝑡) ≥ 𝜑 [ 𝑎 𝑀𝑝(𝑆𝑎, 𝑇𝑐′, 𝑡) + (1 − 𝑎) min {𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡), 𝑀𝑝(𝐵(𝑐′, 𝑎′, 𝑏′), 𝑇𝑐′, 𝑡),𝑀 𝑝 2(𝐴(𝑎, 𝑏, 𝑐), 𝑇𝑐′, 𝑡).𝑀 𝑝 2(𝐵(𝑐′, 𝑎′, 𝑏′), 𝑆𝑎, 𝑡) , 1 2 [𝑀𝑝(𝐴(𝑎, 𝑏, 𝑐), 𝑆𝑎, 𝑡) + 𝑀𝑝(𝐵(𝑐′, 𝑎′ , 𝑏′), 𝑇𝑐′, 𝑡)]} ] 𝑀𝑝(𝑥, 𝑧, 𝑞𝑡) ≥ 𝜑(𝑎 𝑀𝑝(𝑥, 𝑧, 𝑡) + (1 − 𝑎)𝑚𝑖𝑛{1,1, 𝑀𝑝(𝑥, 𝑧, 𝑡), 1} = 𝜑𝑀𝑝(𝑥, 𝑧, 𝑡) > 𝑀𝑝(𝑥, 𝑧, 𝑡) ⟹ 𝑥 = 𝑧 ⇒ 𝑥 = 𝑦 = 𝑧 Now we prove that 𝑆𝑥 = 𝑇𝑥 𝑀𝑃(𝐴(𝑥, 𝑦, 𝑧), 𝐵(𝑦, 𝑧, 𝑥), 𝑞𝑡) ≥ 𝜑 [ 𝑎 𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑡) + (1 − 𝑎) min {𝑀𝑝(𝐴(𝑥, 𝑦, 𝑧), 𝑆𝑥, 𝑡), 𝑀𝑝(𝐵(𝑦, 𝑧, 𝑥), 𝑇𝑦, 𝑡),𝑀 𝑝 2(𝐴(𝑥, 𝑦, 𝑧), 𝑇𝑦, 𝑡).𝑀 𝑝 2(𝐵(𝑦, 𝑧, 𝑥), 𝑆𝑥, 𝑡) , 1 2 [𝑀𝑝(𝐴(𝑥, 𝑦, 𝑧), 𝑆𝑥, 𝑡) + 𝑀𝑝(𝐵(𝑦, 𝑧, 𝑥), 𝑇𝑦, 𝑡)]} ] 𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑞𝑡) ≥ 𝜑(𝑎 𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑡) + (1 − 𝑎)𝑚𝑖𝑛{1,1, 𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑡), 1} = 𝜑𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑡) > 𝑀𝑝(𝑆𝑥, 𝑇𝑦, 𝑡) ⟹ 𝑆𝑥 = 𝑇𝑥 𝑆𝑥 = 𝑇𝑥 = 𝐵(𝑥, 𝑦, 𝑧) = 𝐴(𝑥, 𝑦, 𝑧) or 𝑆𝑥 = 𝑇𝑥 = 𝐵(𝑥, 𝑥, 𝑥) = 𝐴(𝑥, 𝑥, 𝑥) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2389 https://internationalpubls.com Also by condition (ii) we have, 𝑥 = 𝐵(𝑥, 𝑥, 𝑥) Thus 𝐴(𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥 ∎ Example 3.1.1 Let𝑋 = [0,1] with the metric 𝑑 defined by 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| and for each 𝑡 ∈ [0,1], define M(x, y, t) = { t t + |x − y| , if t > 0; 0, if t = 0 for all 𝑥, 𝑦 ∈ 𝑋. Clearly (X, ℱ,∗) be a fuzzy metric space, with 𝑎 ∗ 𝑏 = min{𝑎, 𝑏}. Let 𝑆, 𝑇: 𝑋 → 𝑋and 𝐴, 𝐵:𝑋 × 𝑋 × 𝑋 → 𝑋 defined by A(x, y, z) = 2x + y + z 2 S(x) = { x, if0 ≤ x < 1; 5 2 , ifx ≥ 1. B(x, y, z) = yT(x) = { x, if0 ≤ x < 1; 5, ifx ≥ 1. Also the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc. Clearly all the conditions of the above theorem are satisfied. Also 𝑆𝐴(0,0,0) = 𝐴(𝑆0, 𝑆0, 𝑆0) and 𝑇𝐵(0,0,0) = 𝐵(𝑇0, 𝑇0, 𝑇0) So, (A, S) and (B, T) are owc maps and(0, 0, 0) is the common tripled fixed point of A, B, S and T. The Mesh Diagram for the given example is shown in Fig [3.1]. Fig [3.1] Theorem: 3.2 Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2390 https://internationalpubls.com (i) 𝑀𝑃(𝐴(𝑥, 𝑦, 𝑧), 𝐵(𝑢, 𝑣, 𝑤), 𝑞𝑡) ≥ min { 𝑀𝑝(𝑆𝑥, 𝑇𝑢, 𝑡),𝑀(𝐴(𝑥, 𝑦, 𝑧), 𝑇𝑢, 𝑡).𝑀𝑝−1(𝐵(𝑢, 𝑣, 𝑤), 𝑆𝑥, 𝑡)} for all 𝑥, 𝑦, 𝑧, 𝑢, 𝑣, 𝑤 ∈ 𝑋, 0 ≤ 𝑎 ≤ 1, 𝑝 ≥ 1. (ii) 𝑦 = 𝐵(𝑥, 𝑦, 𝑧) Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that𝐴(𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. Theorem: 3.3Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: 𝑀(𝐴(𝑥, 𝑦, 𝑧), 𝐵(𝑢, 𝑣, 𝑤), 𝑞𝑡) ≥ 𝑚𝑖𝑛 {𝑀(𝑆𝑥, 𝑇𝑢, 𝑡) + 1 2 (1 + 𝑀(𝐴(𝑥, 𝑦, 𝑧), 𝑆𝑥, 𝑡) 𝑀(𝐵(𝑢, 𝑣, 𝑤), 𝑇𝑢, 𝑡) )} for all 𝑥, 𝑦, 𝑢, 𝑣 ∈ 𝑋 (i) 𝑦 = 𝐵(𝑥, 𝑦) Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that 𝐴(𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. References [1] T. G. Bhaskar and V. Lakshmikantham, "Fixed point theorems in partially ordered metric spaces and applications," *Nonlinear Analysis: Theory, Methods & Applications*, vol. 65, no. 7, pp. 1379–1393, 2006. [2] J. X. Fang, "Common fixed point theorems of compatible and weakly compatible maps in Menger spaces," *Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 5–6, pp. 1833–1843, 2009. [3] A. George and P. Veeramani, "On some results in fuzzy metric spaces," *Fuzzy Sets and Systems*, vol. 64, pp. 395–399, 1994. [4] G. 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